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Staircase
Age 16 to 18
Challenge Level 
This problem was suggested by Yatir Halevi.
Solve the equation:
$$x^{(x^3)}=3.$$
Solve the equation:
$$x^{(x^{(x^3)})}=3.$$
Can you find solutions to all equations of the form:
$$x^{x^{x^{x^{x^{x^{...^{a}}}}}}}=a$$
where the sequence of powers is defined in the same way and $a$ is a positive integer?
Explain what you have done and prove that you have found all possible solutions.
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